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EDIT: So after watching the video
Cancerofthehead recommented. The suggestion was rule #6 which says: "Forts that you control AND are in your territory, have a friendly zone of control, additionally to their normal ZoC. You can ALWAYS MOVE INTO OF FROM a province with friendly ZoC. So their fort Unterkarnten is projecting a friendly ZoC on my fort province, while their fort that I was besieging in Etschtal projects a friendly ZoC on the province of Lienz, therefore allowing them to ignore my fort's hostile ZoC thanks to their forts' friendly ZoC.
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Small correction: Friendly ZoC does not cross borders into enemy provinces (the same as hostile ZoC).
So Unterkärnten does NOT project a ZoC (neither hostile nor friendly) on Alta Carinzia and the move would still be possible without a fort in Unterkärnten (because "can always move to adjacent hostile fort when not on a hostile fort" (in some situations you can even move to adjacent hostile forts when on a hostile fort but that does not apply here).
Alta Carnizia -> Lienz is possible because of Eschtal's friendly ZoC, as you correctly noticed.
Unterkärnten's only effect would be the reverse move: Without a fort in Unterkärnten (and no forts in Steiermark, Graz and Krain) the movement Lienz- > Alta Carnizia -> Unterkärnten would not be possible.
How could the move have been prevented?
- Occupying Unterkärnten would not have a big effect, an army coming from Wienerwald -> Steiermark -> Alta Carnizia could still pass to Lienz/Eschtal
- Occupying Eschtal before(!!!) Austria's army started its move: Occupying Eschtal removes the frZoC from Lienz, making any move Alta Carnizia -> Lienz impossible.
As movement orders check for ZoCs only when given it can be possible that Austria bypasses Alta Carnizia even after Eschtal fell (as long as the movement was ordered when Eschtal was still under siege and therfore still projected frZoC on Lienz).
- Owning Lienz (important: "own" =/= "control"): If you owned Lienz (regardless of occupation status) any move Steiermark/Unterkärnten -> Alta Carnizia -> Lienz would be impossible because ZoCs never cross hostile borders and therefore Lienz isn't in Eschtal's frZoC.
However, owning Lienz only prevents movements from East to West, it does not block movements from West to East (Eschtal -> Lienz ("when not in hoZoC can always move to adjacent hoZoC"), Lienz -> Alta Carnizia ("when not on hostile fort can always move to adjacent hostile fort"), Alta Carnizia -> Unterkärnten/Steiermark ("can always move to frZoC")).
That said, this was an easy example for applying fort movement rules ingame. There are much more obscure examples.
Fort rules are complicated and sometimes interact with each other in unexpected (and sometimes unforseeable) ways but usually you can find an (at least theoretical) explanation why an AI move was possible after it happened. (And I said "theorectical" because there is no easy way to check an enemy army's Return Province which can often make big differences and also no easy way to check when and from where an AI movement order was issued)